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Journal of Probability and Statistics
Volume 2017 (2017), Article ID 8510782, 13 pages
https://doi.org/10.1155/2017/8510782
Research Article

A Conditional Fourier-Feynman Transform and Conditional Convolution Product with Change of Scales on a Function Space II

Department of Mathematics, Kyonggi University, Suwon 16227, Republic of Korea

Correspondence should be addressed to Dong Hyun Cho

Received 29 July 2016; Accepted 24 November 2016; Published 28 February 2017

Academic Editor: Ramón M. Rodríguez-Dagnino

Copyright © 2017 Dong Hyun Cho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Using a simple formula for conditional expectations over continuous paths, we will evaluate conditional expectations which are types of analytic conditional Fourier-Feynman transforms and conditional convolution products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the measures on the Borel class of . We will then investigate their relationships. Particularly, we prove that the conditional transform of the conditional convolution product can be expressed by the product of the conditional transforms of each function. Finally we will establish change of scale formulas for the conditional transforms and the conditional convolution products. In these evaluation formulas and change of scale formulas, we use multivariate normal distributions so that the conditioning function does not contain present positions of the paths.

1. Introduction

Let denote an analogue of Wiener space which is the space of real-valued continuous functions on the interval [1]. The conditional expectations which are types of analytic conditional Fourier-Feynman transforms and conditional convolution products on are introduced by the author [2] using a conditioning function which contains present positions of the paths. According to the author’s paper, he investigated the effects that drift has on the conditional Fourier-Feynman transform, the conditional convolution product, and various relationships that occur among them. In particular, he derived several change of scale formulas for the conditional transforms and the conditional convolution products, which simplify evaluating the conditional expectations, because the probability measure used on cannot be scale-invariant [3, 4].

Let be absolutely continuous on and let be of bounded variation with a.e. on . Define a stochastic process by for and , where denotes the Paley-Wiener-Zygmund stochastic integral [1]. For a partition of , define random vectors and by With the conditioning function which contains the present position of the path , the author [5] evaluated the conditional Fourier-Feynman transforms and the conditional convolution products of the functions given bywhere is a complex Borel measure of bounded variation on , with , and is an orthonormal subset of . He then investigated relationships between the conditional Fourier-Feynman transforms and the conditional convolution products of the functions given by (3).

In this paper, using a simple formula for conditional expectations over continuous paths [6], we evaluate conditional expectations of generalized cylinder functions and the functions in a Banach algebra which plays significant roles in Feynman integration theories and quantum mechanics. We then investigate their relationships. Particularly, we prove that the conditional transform of the convolution product can be expressed by the product of the conditional transforms of each function. Finally we establish change of scale formulas for the conditional expectations. In these evaluation formulas and change of scale formulas we use multivariate normal distributions so that the conditioning function does not contain the present positions of the paths. In fact, with the conditioning function which does not contain the present position of the path , we evaluate conditional expectations, that is, the conditional Fourier-Feynman transforms and the conditional convolution products of the functions given by (3). We then show that the -analytic conditional Fourier-Feynman transform of the conditional convolution product for the functions and which are described in (3) can be expressed by the formula for a nonzero real , almost surely , and almost surely , where and is the probability distribution of on the Borel class of . Compared with the results in [5, 7], the conditioning function in this paper does not contain the present position of and the effects of drift depend on the polygonal function of so that we can generalize the theorems in [5, 7] and the results of this paper do not depend on a particular choice of the initial distribution of the paths.

2. Preliminaries on an Analogue of Wiener Space

We begin this section with introducing a concrete form of an analogue of Wiener space which is our underlying space.

For a positive real let denote the space of real-valued continuous functions on the time interval with the supremum norm. For with let be the function given by . For   () in the Borel class of , the subset of is called an interval and let be the set of all such intervals. For a probability measure on , let where for . , the Borel -algebra of , coincides with the smallest -algebra generated by and there exists a unique probability measure on such that for all . This measure is called an analogue of Wiener measure associated with the probability measure [1].

Let and denote the sets of complex numbers and complex numbers with positive real parts, respectively. Let be integrable and let be a random vector on assuming that the value space of is a normed space with the Borel -algebra. Then we have the conditional expectation of given from a well-known probability theory [8]. Furthermore, there exists a integrable complex-valued function on the value space of such that where is the probability distribution of . The function is called the conditional -integral of given and it is also denoted by .

For an extended real number with , suppose that and are related by (possibly if ). Let . For let be a measurable function on such thatThen we write

Let be a partition of , where is a fixed nonnegative integer. Let be of bounded variation with a.e. on . Let be absolutely continuous on and define stochastic processes by for and , where denotes an indicator function. Define a random vector by for . For , let and for any function on , define a polygonal function of byfor . For , define a polygonal function of by (11), where is replaced by . If , is interpreted as with . For and letFor , , , and any nonsingular positive matrix on , letwhere denotes the dot product on . Let be the identity matrix on . For a measurable function , let let , and let for . Suppose that exists, where the expectation is taken over the first variable. By Theorem in [2] and Theorem in [6]for a.e. , where , is the probability distribution of on and the expectation is taken over the variable . Let be the right-hand side of (15). If, for a.e. and a.e. , has an analytic extension on , then it is called a generalized analytic conditional Fourier-Wiener transform of given with the parameter and is denoted by Moreover if has a limit as approaches through , then it is called a generalized -analytic conditional Fourier-Feynman transform of given with the parameter and is denoted by For , define a generalized -analytic conditional Fourier-Feynman transform of given by the formula Let be defined on . For and , redefine and suppose that exists. By Theorem in [2] and Theorem in [6]Let be the right-hand side of (19). If has an analytic extension on , then it is called a generalized conditional convolution product of and given with the parameter and denoted by Moreover if has a limit as approaches through , then it is called a generalized conditional convolution product of and given with the parameter and denoted by

For , let let be the subspace of generated by , and let be the orthogonal complement of . Let be the orthogonal projection given by and let be the orthogonal projection. For , , and any nonsingular positive matrix on , let

The following lemmas are useful to prove the results in the next sections [5].

Lemma 1. Let . Then for a.e. where is the multiplication operator defined by

Lemma 2. Let and . Then where .

Lemma 3. Let be a subset of such that is an independent set. Then the random vector has the multivariate normal distribution with mean vector and covariance matrix . Moreover, for any Borel measurable function one has where means that if either side exists, then both sides exist and they are equal.

Remark 4. Because the Borel sets of are always scale-invariant measurable and we use the Borel class of on which is defined [1], the scale-invariant measurability is not essential in (7).

3. Conditional Fourier-Wiener and Fourier-Feynman Transforms

Let , let be any fixed positive integer, and let be an orthonormal subset of such that both and are independent sets. Let be the space of cylinder functions having the formfor a.e. , where and . Without loss of generality we can take to be Borel measurable.

Theorem 5. Let and let be given by (29). Then for for a.e. and a.e. , where , , andMoreover .

Proof. For let , where . For and a.e. we have by Lemmas 1 and 2where and are given by (12) and (13), respectively. Using the same method as used in the proof of Theorem in [7] We note that if , then by the change of variable theorem and for we have by Schwarz’s inequalityNow, by Morera’s theorem with aids of Hölder’s inequality and the dominated convergence theorem, we have (30) for . Since and by (35), the final result follows by the change of variable theorem and Young’s inequality [9].

Corollary 6. Let and let be given by (29). (1) If , then one has for , and where for (2) If , then one haswhere for and for (3) If and , then one haswhere for and for

By the dominated convergence theorem and Theorem 5 we have the following theorem.

Theorem 7. Let be given by (29). Then for a nonzero real , a.e. , and a.e. , exists and it is given by the right-hand side of (30), where is replaced by . Furthermore .

Using Lemma in [10] we can prove the following theorem.

Theorem 8. Let be given by (29) with , let be a nonzero real number, and let . (1) Suppose that for . Then for a.e. and a.e. , is given by where is given by (31). Furthermore .(2) If , then for a.e. and a.e. , , exists and it is given by the right-hand sides of (38), where is replaced by . Furthermore .(3) If and , then for , is given by the right-hand sides of (41), where is replaced by .

Using the same method as used in the proof of Theorem in [7], we can prove the following theorem.

Theorem 9. Let be given by (29) with , and let . (1)Suppose that for . For a.e. and a.e. , let . Thenfor , and for as approaches through .(2)Suppose that . Let . Then one has (45) if and has (46) if , where is replaced by .(3)Suppose that and . Let . Then one has (45) if and has (46) if , where is replaced by .

4. Relationships between Conditional Fourier-Feynman Transforms and Convolution Products

In this section we evaluate the conditional convolution products and investigate their relationships.

Now, we have the following two theorems by (35) and Theorems and in [7].

Theorem 10. Let , , and and be related by (29), respectively, where . Furthermore let and . Then for , a.e. , and a.e. , exists and is given by Moreover one has if either or , if , and if .

Theorem 11. Let be a nonzero real number. Then for or and a.e. , one has the following:(1)if , then ,(2)if , then ,(3)if and , then ,(4)if and , then ,(5)if and , then .

Theorem 12. Let and let . Then for , a.e. , and a.e. , one has

Proof. We note that is well-defined by Theorems 5 and 10. By those theorems as stated above we have for , a.e. , and a.e. Let and . Then we have by the change of variable theorem and Lemma 2which completes the proof.

We now have the following theorem from Theorems 5, 10, 11, and 12.

Theorem 13. Let be a nonzero real number. Then one has the following: (1)if , then one has for a.e. and a.e. (2)if and , then

5. Evaluation Formulas for the Functions in a Banach Algebra

Let be the function defined bywhere is a complex-valued Borel measure of bounded variation over . For a.e. , let be given by

By Theorem in [11], we can prove the following theorem.

Theorem 14. Let and let be given by (54). Then for , a.e. , and a.e. , is given by